Circles (Part 1–2)

Equation of a circle: (x − a)² + (y − b)² = r²
01 • Equation of a circle: (x − a)² + (y − b)² = r²
tap to expand
Distance from (x, y) to (a, b) equals r.
Read centre & radius from standard form
02 • Read centre & radius from standard form
tap to expand
For x² + y² = r² ⇒ centre (0,0). For (x − a)² + (y − b)² = r² ⇒ centre (a, b).
Equation from centre (5, −4) and radius 3
03 • Equation from centre (5, −4) and radius 3
tap to expand
(x − 5)² + (y + 4)² = 3² ⇒ (x − 5)² + (y + 4)² = 9.
Circle with diameter AB (centre = midpoint)
04 • Circle with diameter AB (centre = midpoint)
tap to expand
Radius = half of AB; centre = midpoint of A and B.
Complete the square to find centre & radius
05 • Complete the square to find centre & radius
tap to expand
Group x & y, complete squares, compare with (x − a)² + (y − b)² = r².
Circle theorems: semicircle 90°, chord bisector, tangent ⟂ radius
06 • Circle theorems: semicircle 90°, chord bisector, tangent ⟂ radius
tap to expand
Facts: angle in semicircle = 90°, radius ⟂ tangent, perpendicular from centre bisects a chord.
Circle through 3 points via perpendicular bisectors
07 • Circle through 3 points via perpendicular bisectors
tap to expand
Centre at intersection of two perpendicular bisectors.
Two circles: non-intersecting & touching cases
08 • Two circles: non-intersecting & touching cases
tap to expand
Use AB vs r₁, r₂: AB>r₁+r₂ (separate), AB=r₁+r₂ (external touch), AB<|r₁−r₂| (one inside), etc.
Two circles: intersecting & common chord
09 • Two circles: intersecting & common chord
tap to expand
Common chord joins intersection points; length via Pythagoras.
Line–circle intersections (solve simultaneously)
10 • Line–circle intersections (solve simultaneously)
tap to expand
Substitute line into circle ⇒ quadratic; two solutions = 2 intersections.
Tangent test: repeated root (x + 2y = 30)
11 • Tangent test: repeated root (x + 2y = 30)
tap to expand
After substitution, discriminant = 0 (repeated root) ⇒ tangent.
One circle inside another (compare centres & radii)
12 • One circle inside another (compare centres & radii)
tap to expand
Centres on x-axis; compare centre distance with |r₁−r₂| to show smaller is inside.